Section 12'3 Measures of Relative Position

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Section 12.3 Measures of Relative Position
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Z-Score

• The z-score for a given data value x is the
  number of standard deviations that x is above or
  below the mean of the data.

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Formula for Z-Score
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Example

• Julio has written two papers in his literature
  class. He scored 87 on the first paper, for
  which the mean of all scores was 82 and the
  standard deviation was 4. He received a 94 on
  the second paper, for which the mean of all
  scores was 91 and the standard deviation was 3.
  In comparison to the other students, did Julio do
  better on the first paper or the second paper?

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Example

• A consumers group tested a sample of 100 soup
  cans. It found that the mean shelf life of the
  cans was 19 months, with a standard deviation of
  2. One particular can from the K-Mart company
  had a z-score of 1.5. What was the shelf life of
  this can?

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Percentile

• A value x is called the pth percentile of a data
  set provided p of the data values are less than
  x.
• Example
• The median height of a plant is 56 centimeters.
  If the 85th percentile for the height of the
  plant is 78 centimeters, find the percent of
  plants that are between 56 and 78 centimeters
  tall.

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Percentile for a given data value

• Given a set of data and a data value x,
• Percentile of score x number of data value
  less than x times 100 / total number of data
  values
• Example
• On an assessment test given to 1200 students,
  Mike's score of 780 was higher than the scores of
  1008 of the students who took the examination.
  What is the percentile for Mike's score?

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Quartiles

• The three numbers Q1, Q2, and Q3 that partition a
  data set into 4 approximately equal portions are
  called the quartiles of the data.
• The median procedure for finding quartiles
• a. Rank the data
• b. Find the median of the data, this is Q2
• c. The Q1 is the median of the data smaller than
  Q2. d. The Q3 is the median of the data
  larger than Q2.

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Box-and-Whisker Plots

• Construction of a Box-and-Whisker Plots
• a. Draw a horizontal scale that extends from the
  minimum data value to the maximum value.
• b. Above the scale, draw a rectangle (box) with
  its left side at Q1 and its right side at Q3.
• c. Draw a vertical line segment across the
  rectangle at the median, Q2.
• d. Draw a horizontal line segment, called
  whisker, that extends from Q1 to the minimum and
  another whisker that extends from Q3 to the
  maximum.